#### Lclm

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\centerline{\fbox{\parbox{5.7cm}{\Large\bf UserFunction Lclm} }}
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\vspace*{1mm}\noindent{\bf Lclm}(u,v:LODO(C,v)).
The arguments are from a non-commutative ring of ordinary differential operators. The meaning of the parameters is as follows.

$C$: Coefficient type.

$v$: Differential variable in $D\equiv\fracsm{d}{dv}$.

\vspace*{1mm}\noindent{\bf Specification.}
A generator for their intersection left-ideal is returned.

\vspace*{1mm}\noindent{\bf Examples.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is given in a separate window.

\4{\tt d1:=D(x)+1/2+1/x;  d2:=D(x)-1/2;}

\4{\tt Lclm(d1,d2|LODO(RATF Q,x)|);}

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The differential operators, $D=\fracsm{d}{dx}$:
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$\displaystyle D+\fracsm{x+2}{2x},\2 D-\fracsm{1}{2}$

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The Lclm:

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$D^2+\fracsm{x+2}{x^2+x}D-\fracsm{x^2+3x+4}{4x^2+4x}$}}

\vspace*{1mm}\noindent{\bf Lclm}(u,v:LDO(C,vs).
The arguments are from a non-commutative ring of partial differential operators.
$C$ determines the coefficient type.

$v=\{v_1,v_2,\ldots\}$ determines the differential operators
$\fracsm{\partial}{\partial v_1},\fracsm{\partial}{\partial v_2},\ldots$.

\vspace*{1mm}\noindent{\bf Specification.}
The generators for their intersection left-ideal are returned.

\vspace*{1mm}{\bf Example.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is given in a separate window.

\4{\tt d1:=D(x)+a/(x+y); d2:=D(y)+a/(x+y);}

\4{\tt Lclm(d1,d2|LDO(RATF Q,\{x,y\},GRLEX)|);}

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The Differential Operators:

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$\partial_x+\fracsm{a}{x+y},\2\partial_y+\fracsm{a}{x+y}$

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The Lclm ideal:

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$\displaystyle <\partial_{xy}+\fracsm{a}{x+y}\partial_x+\fracsm{a}{x+y}\partial_y +\fracsm{a^{2}-a}{x^{2}+2xy+y^{2}}>$

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Term Order: GRLEX, $x>y$ }}

\vspace*{1mm}\noindent{\bf Lclm}$(u,v:LODE(C,y,x)$.
The arguments are linear homogeneous ordinary differential polynomials.
$C:$ Coefficient type.

$y:$ Dependent variable.

$x:$ Independent variable.

\vspace*{1mm}\noindent{\bf Specification.}
The least common left multiple of the arguments $u$ and $v$ is returned.

\vspace*{1mm}\noindent{\bf Example.}
The input is given in {\em Reduce} algebraic mode syntax. The output for each minisession is given in a separate window.

\4{\tt deq1:=Df(y,x)-y/x;  deq2:=Df(y,x)-2*y/x;}

\4{\tt Lclm(deq1,deq2|LODE(RATF Q,y,x)|);}

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\fbox{\parbox{5cm}{

\vspace*{2mm}
The Lclm:

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$y''-\fracsm{2}{x}y'+\fracsm{2}{x^2}y=0$}}

\vspace*{1mm}\noindent{\bf Lclm}(u,v:LDFMOD(C,z,x,O)).
The arguments $u$ and $v$ have the type $LDFMOD$ they represents differential polynomials in the dependent variables $z$ and independnet variables $x$. The meaning of its parameters is as follows.

$C$ determines the type of the coefficients.

$z=\{z_1,z_2,\ldots \}$ denotes the dependent variables in decreasing order.

$x=\{x_1,x_2,\ldots \}$ denotes the independent variables in decreasing order.

$O:LEX|GRLEX$ determines the applied term ordering.

\vspace*{1mm}\noindent{\bf Example.}

\4{\tt
l1:={Df(z,y)+z/y,Df(z,x)+z/x};  l2:={Df(z,y)+z/(x+y),Df(z,x)+z/(x+y)};}

\5{\tt Lclm(l1,l2|LDFMOD(RATF Q,{z},{y,x},GRLEX)|); }

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The Lclm:
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$\displaystyle z_y-\fracsm{x^2}{y^2}z_x-\fracsm{x-y}{y^2},\2 z_{xx}+\fracsm{4x+2y}{x^2+xy}z_x+\fracsm{2}{x^2+xy}z$

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Term Order: $GRLEX,z>y>x$  }}

\vspace*{1mm}\noindent{\bf Related Functions: $Gcrd(u,v)$}

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{\bf References}

D.~Grigoriev, F.~Schwarz, {\em Factoring and Solving Linear Partial Differential Equations}, Computing {\bf 73}, 179-197 (2004).

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